Related papers: A Hint on the External Field Problem for Matrix Mo…
A very elementary model of a single positive hermitian random matrix coupled to an external matrix is defined and studied. Expanding the exact effective action around its classical solution leads to the ``quantum Penner action'', from which…
We give explicit analytic criteria for two problems associated with the Schr\"odinger operator $H = -\Delta + Q$ on $L^2(\R^n)$ where $Q\in D'(\R^n)$ is an arbitrary real- or complex-valued potential. First, we obtain necessary and…
The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a hermitean one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on…
We introduce a dbar-formulation of the orthogonal polynomials on the complex plane, and hence of the related normal matrix model, which is expected to play the same role as the Riemann-Hilbert formalism in the theory of orthogonal…
The partition functions of Hermitian one-matrix models are known to be tau-functions of the KP hierarchy. In this paper we explicitly compute the elements in Sato grassmannian these tau-functions correspond to, and use them to compute the…
We compute the supersymmetric partition function on L(r,1)xS^1, the lens space index, for 4d gauge theories related by supersymmetric dualities and involving non simply-connected groups. This computation is sensitive to the global…
We study a quartic matrix model with partition function $Z=\int d\ M\exp{\rm Tr}\ (-\Delta M^2-\frac{\lambda}{4}M^4)$. The integral is over the space of Hermitian $(\Lambda+1)\times(\Lambda+1)$ matrices, the matrix $\Delta$, which is not a…
We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest O(10) boundstate energies and wave functions to machine precision. The method extends also to Hamiltonians that are…
We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a…
The Hubbard model is a simplified description for the evolution of interacting spin-1/2 fermions on a d-dimensional lattice. In a kinetic scaling limit, the Hubbard model can be associated with a matrix-valued Boltzmann equation, the…
We study the partition function of both Close-Packed Dimers and the Critical Ising Model on a square lattice embedded on a genus two surface. Using numerical and analytical methods we show that the determinants of the Kasteleyn adjacency…
$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- and $q,t$-deformations of the simplest…
We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.
The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…
We suggest model equations, which, from some point of view, describe local interaction of three physical fields: a field of matter, an electromagnetic field and a gravitational field. A base of the model is a field of matter described by…
We numerically study the single-flavor Schwinger model with a topological $\theta$-term, which is practically inaccessible by standard lattice Monte Carlo simulations due to the sign problem. By using numerical methods based on tensor…
We identify the Kontsevich-Penner matrix integral, for finite size $n$, with the isomonodromic tau function of a $3\times 3$ rational connection on the Riemann sphere with $n$ Fuchsian singularities placed in correspondence with the…
A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach…
In this paper, we consider the Neumann problem for a class of Hessian quotient equations involving a gradient term on the right-hand side in Euclidean space. More precisely, we derive the interior gradient estimates for the $(\Lambda,…
In the first part of the talk, I review the applications of loop equations to the matrix models and to 2-dimensional quantum gravity which is defined as their continuum limit. The results concerning multi-loop correlators for low genera and…